In its various reports since 1988, the
Working Group on California Earthquake Probabilities
(WGCEP) has consistently made use of statistical models of time-dependent
earthquake recurrence to help estimate earthquake probabilities in
California. In all WGCEP reports, the need for a larger database of
earthquake recurrence intervals is emphasized. Additional data are
necessary to select from competing statistical models of earthquake
recurrence and to better define the model parameters. The most recent
working group reports (WG99, 1999; WG02, 2002) introduced recurrence intervals
from numerous small and characteristically repeating microearthquake
sequences occurring along the San Andreas fault and, in particular, at
Parkfield (Ellsworth et al., 1999). They included only a small
fraction of the Parkfield data yet that nearly doubled the available
data base from which estimates of the model parameters were made.
However, even this expanded data set was considered less than adequate
by the working groups. At Parkfield a significantly larger set of small repeating
sequences than was used by the working groups exists (184 total sequences
composed of 1073 events and 889 recurrence intervals; compared to 17
Parkfield sequences of 117 events and 100 recurrence intervals used for
the WG model) and many additional sequences and events have recently
been identified by our research group along the central SAF between
Parkfield and the southeast terminus of the Loma Prieta rupture
(Nadeau and McEvilly, 2003). In
total, our group has now identified a total of 515 sequences yielding
2079 recurrence intervals from sequences whose characteristic magnitudes
range from -0.7 to 3.5 (Figure 17.1).

The potential of such a large data set for helping select and refine
time-dependent recurrence models is considerable,
yet serious questions remain regarding the use of small earthquake data
for forecasting large earthquakes (e.g. Is the variance of earthquake
recurrence intervals independent of magnitude, and what normalization (if any)
should be used to account for variations in magnitude among
characteristic sequences?). Nadeau and Johnson (1998) found that the
average recurrence interval of sequences scales reasonably well for
sequence magnitudes ranging from -0.7 to 6 under similar fault
loading conditions. However, they did not analyze the variance of
intervals as a function of magnitude. Ellsworth et al. (1999) analyzed
magnitude dependent variance for the small data used by the working groups, and noted
that the variance for both the small and large sequences was
comparable. However results of their analysis, admittedly based on a small
data set, had a large uncertainty. Adding to the difficulties
in evaluating magnitude dependent variance are problems associated with
spatial and temporal variations of fault loading rates on recurrence
time variance. For example if significant slip rate transients occur
over time scales comparable to the recurrence intervals of sequences
(as has been observed at Parkfield) the variance of the intervals
should be significantly greater than for sequences whose recurrence
times are long compared to the transients.

Previous efforts at variance determinations have also relied on data
dependent normalizations where the intervals for each sequence were
divided by the average recurrence time of the sequences (Figure 17.1, top).
However as pointed out by Matthews et al. (2002), this significantly
under estimates the variance for typical characteristic sequences where
the total number of repetitions are typically small. Matthews et al.
also point out that discrimination between competing probability models
of interval variance for earthquakes is a difficult task, even when
normalization is ignored and relatively large synthetic data sets are used
(e.g. 50 intervals).

The unique attributes of our recurrence interval data set and
auxiliary geodetic measurements of fault loading rates have the potential of
overcoming many of these difficulties and of greatly expanding our
understanding of the time-dependent earthquake recurrence process.
Since our data set is large and since recurrence times scale with
magnitude, our data can also be used to establish a scaling relationship
between average recurrence time and magnitude that is relatively
insensitive to the average intervals of any individual sequence. This
scaling can then be used to normalize the entire recurrence interval
data set in a way that is also insensitive to the average recurrence
intervals. This approach effectively circumvents
the small sample bias problem discussed by Matthews et al. The bottom
panel of Figure 17.1 shows our first attempt at
normalizing our data set in this way. A significantly better fit to a
lognormal-like distribution is observed compared to the distribution
shown in the top panel of Figure 17.1 where
individual sequence average recurrence intervals are used for normalization.
Also of significance is the larger intrinsic uncertainty determined for
our scale normalized data (0.63 compared to 0.46). This result is
entirely consistent with the remarks of Matthews et al., and
supports the potential of the scaled normalization for
overcoming bias problems associated with the
sequence average recurrence interval
normalization approach. The large number of data also
significantly reduces the confidence bounds of
the intrinsic uncertainty estimate.

Our 2079 recurrence intervals significantly exceed the 50 used by
Matthews et al. in their synthetic dataset used for assessing
the practicality of discriminating between
competing recurrence models. This leaves some hope that discrimination
between models may be feasible after all using real earthquakes and a
scale-normalized dataset. Further investigation of the characteristics
of the recurrence-magnitude scaling and normalization will be required
before reliable conclusions can be reached in these regards, but our
initial results do appear promising. The spatial and temporal
characteristics of fault loading rate variations are also exceptionally
well characterized in the regions where most of our repeating sequences
are occurring. This information can be used to remove and/or asses any
bias in the variance of recurrence interval data that variations in
spatial and temporal fault loading rates introduce into their
distributions. Though our dataset contains only small magnitude
events, the "range" in magnitude that it spans is over 4 magnitude
units. This should also allow us to test (at least for small magnitudes)
the validity of the hypothesis that recurrence interval variance is independent
of magnitude, an implicit assumption used by WG99 and WG02.

Finally, an additional attractive feature of our dataset is that it
continues to increase in size as ongoing repeating events (typically repeats for
each of the 515 sequences occur on the order of every few years or
less). Using these ongoing events to continually add to the recurrence
interval data set can not only provide more data for constraining the
forecast model parameters, but it also can be used to test forecast models on real
events. This can be done by making probabilistic forecasts for the
future small and frequently recurring earthquakes using competing
models and then by assessing the success and failure rates of the
various forecasts.

Our future research plans include
integration of the auxiliary fault loading rate
and recurrence interval data as discussed above, augmentation of the
recurrence interval archive with the ongoing small characteristic
events, parameterizing competing forecast models using our recurrence
and other available recurrence interval data, making small
earthquake forecasts using the competing models and comparing the forecasts'
relative success rates.

Figure 17.1:
Normalized recurrence interval distributions from repeat times of 2594
characteristic small earthquakes (-0.8 3.6) occurring along the
central SAF from 15 km southeast to 160 km northwest of Parkfield, CA.
(Top) Histogram and lognormal fit to recurrence intervals normalized by
the average interval for each sequence. (Bottom) Histogram and fit for
data normalized by the empirical scaling relationship of average
recurrence time with magnitude for all the data. No corrections for
spatial or temporal variations in fault slip rates have been applied,
and all sequences, regardless of the number of member events, have been
included. The exceptionally large number of characteristically
repeating small earthquakes makes normalization by the scaling
relationship insensitive to the biases that plague Tav normalized
data. The surprisingly well behaved Ts scaled results and the large
number of intervals available suggest it might be possible to
discriminate between competing time-dependent earthquake recurrence
models and to test the assumption used by the WG99 and WG02
recurrence model subgroup that aperiodicity of earthquake recurrence is scale
independent.

We appreciate support for this project by the USGS NEHRP program through grant numbers
02HQGR0067 and 03HQGR0065, by NSF through award number 9814605, and by
the U.S. Department of Energy (DOE) under contract No. DE-AC03-76SF00098.

Working Group on California earthquake Probabilities (WG99), Earthquake
probabilities in the San Francisco Bay Region: 2000 to 2030-a summary
of findings, U.S. Geol. Surv., Open-File Rept. 99-517, 1999

Working Group on California earthquake Probabilities (WG02), Earthquake
probabilities in the San Francisco Bay Region: 2003 to 2032-a summary
of findings, U.S. Geol. Surv., Open-File Rept. (to be determined), 2002.